For ease of presentation, \(g(u)\)  denotes the test function for an arbitrary property. We denote by \(H_0\)  the underlying null hypothesis and by \(H_a\)  the alternative hypothesis. For example, when we test \(\text{V}^\text{sym}\) , \(g(u)\)  is \(g^v_{i,j,a,b}(u)\) , \(H_0\)  is symmetry in variables, and \(H_a\)  is asymmetry in variables.
To perform the hypothesis test, the realizations of test functions \(g\)  from the data are obtained. Then, the constructed covariance \(\hat{\mathbf {C}}^{H_0}(\mathbf {h},u) = \left( C_{ij}^{H_0}(\mathbf {h},u)\right)_{i,j=1,\ldots ,p}\)  under \(H_0\)  described in Table REF  based on sample estimators \(\hat{C}_{ij}(\mathbf {h},u)\) , \(\hat{\rho }_1(\mathbf {h},u)\) , \(\hat{\rho }_2(\mathbf {h})\) , \(\hat{\rho }_3(u)\) , \(\hat{\rho }_4(\mathbf {h},u)\) , \(\hat{\rho }_5(\mathbf {h},u)\) , or \(\hat{\rho }_{6,ij}(\mathbf {h})\)  of the original data is used to generate two independent reference data sets. We calculate the test functions from these two reference data sets, denoted by \(g^{H_0}\)  and \(g^R\) . The superscript “R” stands for reference. It is noteworthy that there is no difference in obtaining \(g^{H_0}\)  and \(g^R\) , both of which are test functions obtained from simulated data samples with the same covariance under \(H_0\) . We simply use one of them as a reference and denote it by \(g^R\) .
We know that, in principle, \(g^{H_0}\)  and \(g^R\)  are close to zero. Then, the rank-based test for the functional data is used. Suppose there are \(n_g\)  and \(n_{g^{H_0}}\)  curves in \(g\)  and \(g^{H_0}\) , respectively.
For each curve in \(g\) , mix it with all curves in \(g^R\) , and calculate its rank using an increasing order of modified band depths [1]}. Denote all these ranks by \(r_1,r_2,\ldots ,r_{n_g}\) . Do the same for \(g^{H_0}\) , and calculate the rank of each curve in \(g^{H_0}\)  among \(g^R\) , denoted by \(r^{\prime }_1,r^{\prime }_2,\ldots ,r^{\prime }_{n_{g^{H_0}}}\) . Mix \(r_1,r_2,\ldots ,r_{n_g}\)  with \(r^{\prime }_1,r^{\prime }_2,\ldots ,r^{\prime }_{n_{g^{H_0}}}\)  and calculate the ranks of \(r_1,r_2,\ldots ,r_{n_g}\)  in an increasing order, denoted by \(q_1,q_2,\ldots ,q_{n_g}\) . The final test statistic is
\(W=\dfrac{1}{n_g}\times \dfrac{\sum ^{n_g}_{i=1}q_i}{n_g+n_{g^{H_0}}}.\) 
